Pyramid Surface Area Calculator

A pyramid’s surface area depends on its base shape, but this calculator handles that with just one input change. Pick a regular base — triangular, square, pentagonal, hexagonal, or any other regular polygon — and enter only the number of sides and the side length; the calculator works out the base’s apothem and area for you. Or pick a rectangular (non-square) base and enter its length and width directly. Either way, you can supply whichever you know: the pyramid’s vertical height or its slant height.

Every mode reports the base area, the lateral (side) area, and the total surface area, so you can see exactly how the final number was built.

Pyramid Surface Area Formula

Total Surface Area = Base Area + Lateral Area, where Lateral Area = ½ × Base Perimeter × Slant Height

Where:
Base Area = the area of the pyramid’s flat base
Base Perimeter = the distance around the edge of the base
Slant Height = the distance from the base’s edge midpoint up to the apex, measured along a triangular face (not the same as the pyramid’s vertical height)

This general formula holds for any pyramid with a regular base — only the base-area and base-perimeter formulas change depending on how many sides the base has, which is why this calculator lets you set the number of sides directly.

Regular Pyramids (Triangular, Square, Pentagonal, Hexagonal…)

Finding the Base Area and Apothem from Side Length

Base Area = (n × Side²) ÷ (4 × tan(π/n)), where n is the number of base sides

Apothem = Side ÷ (2 × tan(π/n))

The apothem is the perpendicular distance from the base’s center to the midpoint of one of its edges. Rather than asking you to measure or look up the apothem separately, the calculator derives it from the number of sides and the side length alone — the same one formula covers a triangular base (n = 3), a square base (n = 4), a pentagonal base (n = 5), a hexagonal base (n = 6), or any other regular-polygon base.

Slant Height from Either the Height or the Slant Height Itself

Slant Height = √(Height² + Apothem²)

If you already know the slant height, you can enter it directly and skip this step. If you only know the pyramid’s vertical height, the calculator finds the slant height using the right triangle formed by the height, the apothem, and the slant height itself.

Worked Example: A Square Pyramid

A square pyramid with 4 sides, a 10-ft base side length, and a 12-ft height: Apothem = 10 ÷ (2 × tan(45°)) = 5 ft, Base Area = (4 × 10²) ÷ (4 × tan(45°)) = 100 square feet, Perimeter = 4 × 10 = 40 feet, Slant Height = √(12² + 5²) = √169 = 13 ft, Lateral Area = 0.5 × 40 × 13 = 260 square feet, Total SA = 100 + 260 = 360 square feet.

Worked Example: A Triangular Pyramid (Given the Slant Height Directly)

A triangular pyramid with 3 sides, a 6-ft base side length, and an 8-ft slant height already known: Apothem = 6 ÷ (2 × tan(60°)) ≈ 1.73 ft, Base Area = (3 × 6²) ÷ (4 × tan(60°)) ≈ 15.59 square feet, Perimeter = 3 × 6 = 18 feet, Lateral Area = 0.5 × 18 × 8 = 72 square feet, Total SA = 15.59 + 72 ≈ 87.59 square feet.

Rectangular-Base Pyramids

Total Surface Area = (Length × Width) + Length×√(Height²+(Width/2)²) + Width×√(Height²+(Length/2)²)

A rectangular base (where the length and width aren’t equal) has two different pairs of triangular faces, each with its own slant height — one running up from the longer edges, one from the shorter edges — so the calculator finds both separately before adding everything together.

Worked Example

A rectangular-base pyramid 8 ft long, 6 ft wide, and 10 ft tall: Base Area = 8 × 6 = 48 square feet, Slant height (up the 6-ft sides) = √(10² + (8/2)²) = √116 ≈ 10.77 ft, Slant height (up the 8-ft sides) = √(10² + (6/2)²) = √109 ≈ 10.44 ft, Lateral Area = (8 × 10.44) + (6 × 10.77) ≈ 148.14 square feet, Total SA = 48 + 148.14 ≈ 196.14 square feet.

Height vs. Slant Height — What’s the Difference?

A pyramid’s height is the straight vertical distance from the apex down to the base’s center — the measurement you’d get dropping a plumb line. The slant height is different: it’s the distance from the apex down to the midpoint of one base edge, measured along the tilted surface of a triangular face. Because the apex, the base’s center, and an edge’s midpoint form a right triangle, the two are related by the Pythagorean theorem — which is exactly how this calculator converts between them when you supply only one.

Frequently Asked Questions

How do I find a triangular pyramid’s surface area from just the side length?

Set the number of base sides to 3, enter the base side length, and supply either the pyramid’s height or its slant height. The calculator derives the base’s apothem and area automatically, so no separate apothem lookup is needed.

What’s the surface area formula for a square pyramid?

Set the number of base sides to 4. The calculator uses Total SA = Base Area + Lateral Area, where Base Area = (4 × Side²) ÷ (4 × tan(45°)) and Lateral Area = ½ × Perimeter × Slant Height.

Does this work for a rectangular (non-square) base pyramid?

Yes — switch to the Rectangular Base mode and enter the length, width, and height directly. This mode is separate from the Regular Pyramid mode because a rectangular base’s two pairs of triangular faces have different slant heights.

How do I find the slant height if I only know the height?

For a regular pyramid, the calculator finds it automatically from Slant Height = √(Height² + Apothem²) once you enter the height. For a rectangular base, it computes both of the two distinct slant heights the same way, using each side’s half-width in place of the apothem.